- Email: [email protected]
Bounded quadratic systems in the plane
JOURNAL
OF DIFFERENTIAL
EQUATIONS
7, 25 l-273
Bounded Quadratic R. J. DICKSON
(1970)
Systems in the Plane* AND L.
Lockheed Palo Alto Research Laboratory, Received November
M.
PERKO
Palo Alto, California
94304
8. 1968
This paper contains a study of those two-dimensional autonomous systems with quadratic polynomial right-hand sides which have all of their trajectories bounded for t > 0. Such systems will be referred to as bounded quadratic systems. It follows from the PoincarbBendixson Theorem that any such system will have a rest point in the plane and by translating the origin to that rest point it will have the form if = Ax -tf&),
XEE2,
(1)
where A is a constant matrix and where the components of fs(~) are homogeneous quadratic polynomials in x = (x1 , x2). It will be assumed that f2(x) + 0 since linear systems can be integrated in terms of elementary functions. The survey paper of Coppel [I] contains most of the important results for quadratic systems in the plane. At the end of his paper, Coppel states that what remains to be done for quadratic systems is to determine all possible phase portraits and, ideally, to characterize them by means of algebraic inequalities on the coefficients. This paper first establishes necessary and sufficient conditions for a two-dimensional quadratic system to have all of its trajectories bounded for t > 0 and then determines all possible phase portraits for such bounded quadratic systems in the plane. In determining all possible phase portraits for bounded quadratic systems, some characterization by means of algebraic inequalities on the coefficients of system (1)-or (1) under a suitable linear transformation of coordinates-is also accomplished. In order to complete the algebraic characterization, there remains the problem of determining algebraic inequalities on the coefficients which decide the number and stability properties of limit cycles around each isolated rest point. This remains the outstanding unsolved problem for bounded quadratic systems in the plane. The limit cycle structure to be found in bounded plane systems * This work was supported by the Air Force Office of Scientific Research under contract AF 49(638)-1685 and by the Lockheed Independent Research Program. 251 505/7/z-4
252
DICKSON AND PERK0
can be investigated by using results known for general quadratic systems, (cf. [I], pp. 295-298) and by applying the theory of rotated vector fields [2]. This has been carried out for certain (unbounded) quadratic systems in the plane by Yeh Yen-Chien [3] together with an investigation of the uniqueness of limit cycles. While this approach is quite promising, our results are not not sufficiently complete to justify their exposition here.
1.
CLASSIFICATION
OF ROUNDED QUADRATIC
SYSTEMS
Markus has classified the homogeneous quadratic systems in the plane up to affine transformation by classifying the related real linear algebras; (cf. Theorems 6-8 of [Kj). Th e h omogeneous quadratic systems corresponding to the algebras in Theorems 7 and 8 and to those in Theorem 6, Cases (3), (4), (6), (8), and (10) with k y) -& all have ray solutions; i.e., the related algebra has an idempotent element. Now if the homogeneous quadratic system
has a ray solution then the quadratic system (1) will have an unbounded solution; (cf. Theorem 2, $3 in [.5]). Th us, if fi(x) f 0, the system (1) has an unbounded solution unless the quadratic part of the right-hand side is linearly equivalent to one of the following forms (corresponding to the algebras of Theorem 6, Cases (2), (5), (7), (9), and (10) for k < -4, the last two cases corresponding to the last form below):
0. iXl% 11 .X22 i 0 1;
(A) w (Cl
i
03
( -x1x2
Xl%! -I- x22 1; q2
X22 + L-x221 ’
ICI
(2.
First let us show that the system (1) withf,(x) given in (C) has an unbounded trajectory (even though the corresponding homogeneous quadratic system has no ray solution). LEMMA
1.
The system 9 = Ax
+ (
%X2 +x22 , > x22
x(0)
= xg
has an unbounded trajectory (as t -+ co) for somex0 E E2.
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
253
PLANE
Proof. Let D,, = {(x 1 , x2) : xl 3 01,x2 > /I} where 01,fl > 0. Since the only case when the rest points of (2) are not finite in number is when xs = --al, is a line of rest points (i.e., when us1 = 0 and either a,, = 0 or a,, = ass), it follows that for any given matrix A, 01and p can be chosen sufficiently large that D,, contains no rest points of (2). On x1 = CL,
which is positive for all 01> 0 and xs > i3 provided /I is sufficiently large. On x2 = 8, 3i2
=
+ w
a2121
+
a22h
which for u21 > 0 is positive for all /3 > 0 and x1 2 01provided 01is sufficiently large. For u21 = 0 the above expression for 3i’,is positive for ,8 > 0 sufficiently large. Thus, for usI > 0 and the parameters 01and /I sufficiently large, Dae is a positively invariant set which contains no rest points of (2) and therefore the system (2) has an unbounded trajectory (as t 4 00) for some x,, E D,, . This follows, for example, from Hartman’s Theorem 1.1, (p. 202 in [6]). For u21 < 0 consider the curve V(xl , x2) ZG -x1 + kzz2 = 0 for x2 > 0. The flow defined by (2) is upward across V(X, , x2) = 0 for 0 < k < l/(2 ) u2r I) and x2 sufficiently Iarge. This follows since from (2) and the definition of Wl Px2) dV = -2, dt
+ 2kx2d2 = k[l + 2a,,k] x23 + 0(x,2)
> 0
a21I) and x2 sufficiently large. Let
onV(x,,x,)=OforO
D,, = {(x1 7x2) : x1 >, 01,x2 > (xJk)“2}. Then for a,, 0 sufficiently large D,, is a positively invariant set which contains no rest points of (2) and it therefore follows from Theorem 1.1 (p. 202 in [6]) that D,, contains an unbounded trajectory of (2) as t -+ co. This completes the proof of Lemma 1. We next consider Eq. (1) with the quadratic part, f2(x), given in (A). LEMMA
2.
The system
ji=klx+
0
( 1)
x(0) E z?
XIX2
has all of its trajectories bounded (for t > 0) if and only if uI2 = 0, a,, < 0 and a22 < 0. Proof. The critical points at infinity for (3) can be represented by the pairs of diametrically opposite points PI( &l, 0,O) and Pz(O, &l, 0) on the
254
DICKSON
AND
PERK0
equator of the unit sphere in E3 (cf. e.g., Lefschetz [?‘I, p. 201). In terms of local coordinates (x1 , x8) at the point P, , Eq. (3) has the form
For ula f 0 this system has an elliptic sector at (0, 0). This follows, for example, from Theorem 66 on p. 297 of Andronov and Leontovich [S], which is stated in the appendix of this paper. This implies the existence of unbounded trajectories for (3) as t + co in cast a,, + 0. For ai2 =: 0, Eq. (3) is integrable. We obtain for ata _= 0, a,, :z 0 \
x1(t) = Xl(O) &It x2(t) = x,(O) exp [a&
:- F
+‘I
+ xl(O) azl ‘t exp a,,(t I ‘01
Xl(O) - t’) f a,,t’ f- __ (e”“t _ e”“t’ )]df’ alI 1. (4)
If a,, > 0 and ~~(0) $0 then j sr(t)i - w as t--f co. If a,, < 0, xl(t) as t-p COand it follows in this case that / x2(t)l -
M h I x2(O)/ + I x,(O) a,,/a,,
0
i
as t + Co if a 22 < 0 and that 1q(t)! --f co if a2a > 0, X%(O)f 0 and ~~(0) = 0. The remaining case when ur2 = all = 0 is integrable and obviously has unbounded solutions (as t---f co) for ~~(0) sufficiently large. This completes the proof of Lemma 2. Let us next consider the system (1) withf,(x) given in (B). LEMMA
3.
The system
*==A+ EE2 c-Q2 01’ x(0)
(5)
has all of its trajectories bounded (for t > 0) if and only if uzl = 0, a,, .< 0, u22 ,( 0 and alI + az2 < 0. Proof. The system (5) by the pair of diametrically unit sphere in Es. In terms form 32, = -a21x3 2, =
x3(a$3
has only one critical point at infinity represented opposite points Pr( & 1, 0, 0) on the equator of the of local coordinates (xZ , x3) at Pr , Eq. (5) has the + (all -- ad x2x3 + +23222x3+ xa3 +
%$Zx3
+
%“)*
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
255
PLANE
For a2r f 0 this system has a node at (0,O). This follows, for example, from Theorem 66 of [8]. The existence of a node at infinity for a system of even parity such as (5) implies the existence of unbounded trajectories (as t + UJ). For u2i = 0, Eq. (5) is integrable. We find that for a,, = 0
xl(t) = xl(0)enllt
alzxz(o) [exp[(apz - a,&] - I] if a,, f
+ eallt
u22 -
aI,
uz2
if a,, -= uzgI/
/ %2%(O) t ’ x2’(O) + eallt 2u,, - a,, [expP22 r %2(o) t
- u,Jt] - I] if a,, f
2u,, ‘1 ’
if a11 == 2a9,, -‘. !I
I ” (6)
x2(t) = x2(O) eazzt
Thus, when usi = 0, it follows that if a,, < 0 and u22 < 0 then xi(t) and x,(t)--tOast--tco;ifu,,>Oandx,(O)fOthenjx,(t)(~cciast--,co; if a,, > 0, x2(O) = 0 and x,(O) # 0 then / xl(t)\ - co as t - co; if al1 -z 0, us2 < 0 then x2(t) + 0 and xl(t) --f x,(O) - u,,~,(O)/u~~ - .zs2(0)/2a,, ; if U 22 = 0, a,, < 0 then x2(t) = x2(O) and xl(t) +- -~~~x~(O)/u,r -- x,2(0)/u,, ; and finally, if u,r = uz2 = 0 and xz(0)(x2(O) + czr2)f 0 then 1x,(t)\ + co as t + co. This completes the proof of Lemma 3. Lastly, we consider the system (1) with &(x) given in (D). LEMMA
4.
The system Le=llAx+
X22 i -x1x2
+ cx2 2'i
x(0) E E2
(7)
has all of its trajectories bounded (for t > 0) if and only if j c 1 < 2 and one of the following sets of conditions: (i) a,, < 0; (ii) a,, = 0 and u21 = 0; OY (iii) a,, = 0, %l# 09 a12 + a21 = 0 and cu,, + u22 < 0 is sutis$ed. Proof. If 1c / > 2, the homogeneous quadratic system corresponding to (7) has a ray solution (Theorem 6 in [d]) and it follows from Theorem 2, $3 in [5] that (7) has an unbounded solution. If ( c / < 2 then (7) has only one critical point at infinity represented by the pair of diametrically opposite points P,(fl, 0,O) on the unit sphere in l?. In terms of local coordinates (x2 ) xa) at PI , Eq. (7) has the form
2, = --x2 + u&c3 - (Ull - uz2) x2x3 + cx22 - u12x2%3- x23 . 1 3i’, = -x3(ullx3 + u12X2X3 +x22)
(8)
It follows from Bendixson’s theorem, (cf. Lefschetz [7], p. 24), that the above system (8) has a node, a saddle or a saddle-node (two hyperbolic sectors and a fan) at (0, 0), according to whether the index at (0,O) is respectively
256
DIC’KSON
AND
PERK0
+I, -1, or 0. Theorem 65 in Andronov and Leontovich [8], p. 379, which is stated in the appendix of this paper, is used to obtain the following results: If all f 0, then (8) has a saddle-node at (0, 0); if a,, =- 0 and a,,(ar, + CZ& > 0, then (8) has a node at (0,O); if a,, = 0 and ~,,(a~, + a,,) < 0 then (8) has a saddle at (0,O); if a,, = 0, ula + a,, - 0 and u,,(cu,, + uaa)f 0, then (8) has a saddle-node at (0,O). The existence of a node or saddle at infinity for a system of even parity such as (7) implies the existence of an unbounded trajectory (as t + co). Hence, if a,, = 0 and a,,(~,, + a,J f 0, (7) has an unbounded trajectory (as t--t CD). The behavior of (7) for / .x / > 1; i.e., near the equator of the PoincarC sphere as determined by (8) is shown in Fig. 1 for a,, < 0 and also for the case ai1 = 0, ui2 + u2i = 0 and a~,(caal + uaa) < 0; it is shown in Fig. 2 for
FIG.
1
FIG. 2 all > 0 and also for the case a,, = 0, ura + u2r = 0 and u~,(cu,, + uz2) > 0. It follows from the local behavior of (7) near the critical point PI at infinity that all trajectories of (7) are bounded (for t 3 0) if a,, < 0 or if urr = 0, a,, + us1 = 0 and &(~a,, + u2a) < 0 and that (7) has unbounded trajectories (ast-+co)ifu,,>Oorifu,, =0, aI2 + uzl = 0 and a~i(ca,r + u2J > 0. If a,, = us1 = 0 then (7) has x2 as a common factor (i.e., x2 = 0 is a line
BOUNDED QUADRATIC SYSTEMS IN THE PLANE
257
of rest points) and the global behavior of (7) follows from the related linear system $1 =
012 +x2
R, = a22 -
(9)
I*
x1+cx2
This system has a center or a focus at (aZ2- ca12, -al,) according to whether c is equal to zero or not. It follows that with a,, = azl = 0 and / x,, 1 sufficiently large, all trajectories of (7) approach the line of rest points x2 = 0 as t---f co. If a,, = 0, azl f 0 and aI2 + aZ1= caal + a22 = 0 then (7) has (x2 + a,,) as a common factor (i.e., x2 = -alp is a line of rest points) and the global behavior of (7) follows from the related linear system
This system has a focus or center at (0,O) according to whether c is equal to zero or not. It follows that with al, = 0, azl f 0, a,, + uzl = capI + az2 == 0 and / x,, 1 sufficiently large, all trajectories of (7) approach the line of rest points x2 = -aI2 as t -+ co. This completes the proof of Lemma 4. The above results are summarized in the following theorem. THEOREM 1. The quadratic system (1) (with f2(x) + 0) has al2 of its trajectories bounded for t > 0 if and only tf there exists a linear transformation which reduces it to one of the systems (3), (5), or (7) satisfying the conditions of Lemmas 2, 3, or 4, respectively.
2.
CENTERS FOR BOUNDED
QUADRATIC
SYSTEMS
Necessary and sufficient conditions for the existence of a center for a quadratic system in the plane are given in the survey paper of Coppel [I], p. 295. These results are used to determine necessary and sufficient conditions for the existence of a center for systems (3), (5), and (7). A system of the form (1) has a center at (0,O) only if the origin is a center for the corresponding linear system; i.e., only if a11a22- a,,~,, > 0 and all + a22 = 0. This implies that apI # 0 and therefore
is a nonsingular matrix with the property that B-lAB
= [_ol
;I.
258
DICKSON AND PERK0
With x = By, Eq. (1) becomes 9 = B-lABy
+ B-$(By)
which is equivalent to Eq. (3), p. 295 in [I]. The algebraic conditions of [I], p. 295 which are necessary and sufficient for the above system to have a center at (0,O) then yield the following results for the systems (3), (5), and (7), respectively: LEMMA 5.
The system (3) has a center at (0,O) if and only ;f a11 = as” = 0
LEMMA 6.
a@12 < 0.
The system(5) has a center at (0,O) if and onb if w22
LEMMA 7.
and
- a12a2, > 0
and
aI1 +
~2~
= 0.
The system (7) has a center at (0, 0) ;f and only ;f c = a,, = az2 = 0
and
a12az1< 0.
The above results will be used in the next section devoted to determining all possible phase-portraits for bounded quadratic systems.
3. PHASE-PORTRAITS FOR BOUNDED QUADRATIC SYSTEMS Markus [9] has shown that in the plane two Cl systems with isolated critical points and no limit separatrices are equivalent if and only if their separatrix configurations are equivalent. Thus, in case (1) has only a finite number of rest points, it suffices to determine all possible separatrix configurations for (I) in order to determine all possible phase portraits for (1). All possible separatrix configurations for the bounded cases of (1) with only a finite number of rest oints are determined in this section. On the other hand, if (1) does not have a %nite number of rest points, it has a line or curve of rest points and the global behavior is determined by a related linear system. In this way, all possible phase portraits for bounded quadratic systems in the plane are determined in this section. Tung Chin-chu [IO] showed that each limit cycle of (1) contains exactly one rest point in its interior (cf. Theorem 2, p. 296 in [I]). The number of limit cycles around each isolated rest point of (3), (5), or (7) will not be taken up in this paper, since only partial results are available at this time. It is therefore necessary to use the symbols --t-O and 4 in order to exhibit all possible phase-portraits for bounded quadratic systems in the plane. The
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
259
first symbol denotes either a stable node or focus, or a stable or unstable focus on the interior of one or more limit cycles, the outermost of which is externally stable. The second symbol above is similarly defined, the words stable and unstable being interchanged in making the definition. Theorem 6 on p. 299 in [I] which states that a critical point on the interior of a closed path must be either a focus or a center is implicitly used in deriving the results in this section. Let us first consider the system (7) under the conditions of Lemma 4. This is the most interesting case of a bounded quadratic system since the bounded cases of (3) and (5) are integrable. The critical point at infinity P,(&l, 0, 0) for system (7) with j c j < 2 has been discussed in the proof of Lemma 4. Regarding the finite critical points of (7), we have the following lemma. Let d = u1ra2a- u12u21and b = aI3 - a,, + cu,, . LEMMA 8. If a,, f 0, then (7) h as: thyee (jinite) rest points i# d f 0 and b2 > 4d; two (finz’te) rest points ajf b f 0 and either b2 = 4d OY d = 0; one (finite) rest point 23 either b2 < 4d or b = d = 0. In the cuse of three (Jinite) rest points, Qi(xli, xzi) i = 1, 2, 3, it follows that (with proper indexing): x21 < xz2 < x,~; Q2 is a saddle, and Ql and Q3 are either nodes oy foci; Q3 lies to the right (kft) of the line QlQ2 ifull < 0 (a,, > 0).
Proof.
If a,, f 0 then Ax+(
-x,x,
x22 )=o + cx2”
has the solutions x1 = x2 = 0 and x1* = -
(a12 + X2*) x2* %I
with x2* given by x2
* _ -b -
i
(b2 - 4d)1/2
2
The results on the number of (finite) rest points follow immediately. It also follows from the above equations for the rest points that x2- < 0 < x2+ iff d < 0, i.e., iff the origin is a saddle; that x2- < x2+ < 0 iff d > 0 and b > 0; and that 0 < x2- < x2+ iff d > 0 and b < 0. This implies the results stated for the case of three rest points (d f 0, b2 > 4d): First that the x,-coordinates of the rest points are ordered and then (by translating the origin to each of the three rest points, a transformation which leaves the form of the equation as well as the coefficient a,, invariant) that the middle rest point (in the x,-sense) must be a saddle and the upper and lower rest points
260
DICKSON
AND
PERKO
must be nodes or foci. This follows since the above results imply that the coefficient matrix of the linear terms at the upper and lower rest points must have a positive determinant and since these rest points cannot be centers for (7) with aI, f 0 according to Lemma 7. --The fact that ,O+ lies to the right of the line -_lrOO- when a,, < 0 follows from the fact that with the saddle at 0 the line Q+Q- intersects the positive X, axis; i.e., the equation of the line Q+,O- ,
with x2 = 0 yields (with a,, < 0 and d < 0) Xl =
x2+x1x2
-- x2-x1+ + -
x2-
> 0.
This follows since x2+ > x2- and since the above equations for x1* imply that x2+x1- - x2-x1+ > 0 for aI, < 0 and d < 0. (The results for aI, > 0 are proved in an analogous manner.) This completes the proof of Lemma 8. Regarding the finite critical points of (7) in the bounded cases of (7) not covered in Lemma 8, we have one critical point at the origin if 0 and cazl + az8 < 0 and a line of rest points if a11 = 62 + 41 = 0, uzl f a,, = u21 = 0 or if aI, = aI, -+ uzl -= cuzl + uz2 = 0. We now determine all possible separatrix configurations in the case of three finite rest points for the bounded casesof system (7); i.e., for / c 1 < 2, a,, < 0, d f 0 and b2 > 4d. It is no restriction to assume that the saddle is at the origin since translating the origin to any rest point of (7) leaves the form of the equation as well as alI invariant; i.e., we assume that d < 0. Then it is no restriction to further assume that a21< 0 since the transformation x1 -9 x1 , x2 + -x2 transforms (7) into
R, = --a,,~,
+ az2x2 - xlxz - cxz2
which leaves the form of Eq. (7) as well as a,, and d invariant. Note, however, that this transformation is not orientation preserving. In stating the following lemmas, we use the term 0-homeomorphism for an orientation preserving homeomorphism. LEMMA
9.
The separatrix
con$gz/ration fog the system (7) with ) c / < 2,
a,, < 0, azl = 0, and d < 0 is 0-homeomorphic to one of the con$gurations shown in Fig. 3.
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
261
FIG. 3 Proof. It was shown in the proof of Lemma 4 that for 1c 1 < 2 and a,, < 0 the only critical point at infinity is the saddle-node at P,(f 1, 0,O). The behavior near infinity, i.e., near the equator of the PoincarC sphere, as determined by Eq. (8) is shown in Fig, 1 for this case. Note that the equator consists of separatrices. If us1 = 0 and a,, < 0 the xi-axis is composed of trajectories with k1 < 0 for xi > 0 and f, > 0 for x1 < 0. If d < 0, the x,-axis consists of three separatrices (including the origin) with (0,O) as their w-limit set. If d < 0 there are two separatrices (besides the origin) with the saddle at the origin as their a-limit set, one in the upper half plane, Ti , and one in the lower half plane, T, . If d < 0 there is one rest point in the upper half plane and one in the lower half plane (Lemma 8). The PoincarCBendixson Theorem then implies that either the rest point in the upper half plane or an externally-stable limit cycle around that rest point is the w-limit set of Tr and that the rest point in the lower half plane or an externally-stable limit cycle around that rest point is the w-limit set of T, . This completes the proof of Lemma 9, the only possible separatrix configurations for this case being 0-homeomorphic to one of those in Fig. 3.
LEMMA 10.
The separatrix conjiguration for the system (7) with ( c [ < 2, a,, < 0, a,, < 0 and d < 0 is 0-homeomorphic to one of the configurations shown in Fig. 4. Proof. It follows from Lemma 8 that under the given hypothesis, (7) has three (finite) rest points; one at the origin 0, one in the upper half plane Q1 and one in the lower half plane Qs to the right of the line OQ1 . According to the result of Tung Chin-chu given on p. - 296 of - [I], the flow on the line @, is in the same sense on the segments co0 and Qlco and in the opposite sense on the segment OQ1 . Similar results follow for the flow on 0% (cf. Fig. 5). The flow on the x,-axis for u2i < 0 is clockwise (except at x1 = 0).
262
DICKSON
ASD
(a)
PERK0
(b)
(d) FIG. 4
FIG. 5
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
263
PLANE
_The x,-axis and the lines OQ1, OQa divide s” into six open sectors Ri , i = l,..., 6 numbered with increasing argument, R, being in the upper half plane and having the positive x,-axis as part of its boundary. The flow on the x,-axis and on the lines OQ1 and OQa near the origin establishes the fact that (with d < 0) the two separatrices which have the saddle at the origin as their w-limit set, T1 and T, , approach (0,O) as t + co in the sectors R, and R, , respectively; and that the two separatrices which have (0,O) as their a-limit set, T3 and T4, approach (0,O) as t - --cc, in the sectors R, and R, , respectively. It follows from Eq. (8) d escribing the behavior near P, that for aa < 0 the separatrix T5 with PI as its a-limit set approaches PI as t -+ -CC in sector R, (cf. Fig. 5). Now since the boundary of R, as a subset of S2 consists of points of egress (the lines OP, and OW), a trajectory (on the equator), a saddle at 0 and a saddle-node PI , it follows from the Poincart-Bendixson Theorem that the saddle-node PI must be the a-limit set of T, as is shown in Fig. 5. Similarly, considering the boundary of R5 v R, , it follows that Qa or an externally stable limit cycle around Qa must be the w-limit set of the separatrices T4 and T, as is shown in Fig. 5. Finally, considering the boundary of R, V R, u R, u R, , it follows from the Poincare-Bendixson Theorem that the separatrix T3 must have one of the following sets for its w-limit set: (i) the rest point Qr or an externally-stable limit-cycle about Q1 , in which case Tl has Pl( - 1, 0,O) as its a-limit set, as in Fig. 4(a), (ii) the saddle at (0,O); i.e., T, coincides with T, and is a separatrixcycle which contains the rest point Q1 (cf. Theorem 2, p. 296 in [I]) as well as any limit cycle around Q1 , as in Figs. 4(b) and (c) or (iii) the rest point Qa or an externally-stable limit-cycle around Qs , in which case Ti has Qi or an externally-unstable limit-cycle around Q1 as its a-limit set, as in Fig. 4(d). This completes the proof of Lemma 10. Let 7 be the trace of the coefficient to the upper rest point (xi+, ~a+).
matrix
with
the origin
translated
(It is easily shown that 7 = a,, + ua2 + 2cx,+ - xi+.) Regarding the variation of the separatrix configuration in Lemma 10 with respect to a variation of the parameter 7, it is conjectured that we have one of two possible types of behavior, depending on the path in the parameter space (a11 > $2 7 a217 a22 9 c) that corresponds to the variation of 7.
Conjecture. As the parameter 7 increases continuously negative to a large positive value, the separatrix configuration
from a large for the system
264
DICKSON AND PERK0
(7) satisfying the hypotheses of Lemma 10 undergoes one of two possible continuous deformations starting with the configuration 4(a), the upper rest point being a stable node (with no limit-cycle around it): either (i) we have the configuration 4(a) for 7 6: 0 with no limit cycle around the upper rest point, the upper rest point changing from a stable node to a stable focus as T increases through negative values; as T becomes positive, a stable limit-cycle is generated at the upper rest point in 4(a), containing the upper rest point as an unstable focus; as T increases, the limit cycle expands until it intersects the saddle point at the origin and becomes a separatrix cycle at some critical value T* > 0 and we have the configuration 4(c) with no limit cycle around the unstable focus at the upper rest point; finally, as 7 increases through values greater than T*, we have the configuration in 4(d) with no limit cycle around the upper rest point, the upper rest point being first an unstable focus and then becoming an unstable node as T increases without bound; or (ii) we have the configuration 4(a) for 7 less than some critical T* < 0 at which value the configuration 4(b) occurs, the upper rest point going from a stable node to a stable focus as r increases through negative values T << T* and there is no limit cycle around the upper rest point for T .< 0, we have the configuration 4(d) with no limit cycle around the upper rest point, the upper rest point being first an unstable focus and then becoming an unstable node as 7 increases without bound. Which of the two possible variations take place depends on the sign of the critical value T*. (I m pl icr‘t in the conjecture is that T* f 0.) We believe that a proof of this conjecture can be constructed based on Duff’s theory of rotated vector fields [2]; however, certain problems in constructing such a proof such as proving the uniqueness of the limit cycle around the upper rest point have not been dealt with. Many of the details in contructing such a proof would be similar to those contained in the work of Yeh Yen-Chien [3] for a system different from (7) but having a similar separatrix structure. Regarding the lower rest point, during the parameter variation described above, it is conjectured that it is either a stable node or focus with no limitcycle around it or an unstable focus in the interior of a unique stable limit-cycle contained in the interior of R, u R, (defined in the proof of Lemma 10). Numerical examples of each of the configurations in Fig. 4 with at most one cycle around the upper and at most one limit-cycle around the lower
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
265
rest point (including the case with one cycle around each of these rest points) and also examples of the two possible variations of the separatrix configuration with the parameter 7 described in the above conjecture have been obtained on an analog computer. We next determine all possible separatrix configurations in the case of two (finite) rest points for the bounded cases of system (7); i.e., for 1c [ < 2, a,, < 0, b f 0 and either d = 0 or b2 = 4d. We may assume that d = 0, for if d f 0 and b2 = 4d then translating the origin to the second rest point yields a system of the same form as (7) with aI, < 0 left invariant and with the determinant of the transformed matrix A equal to zero. We also note that d = 0 and / c j < 2 imply that b f 0. The same type of argument used in Lemma IO can be employed in this case; however, it is more straightforward to observe that due to the continuity of solutions with respect to the parameter d, letting d + 0 through negative values will result in a continuous deformation of the separatrix configurations in Figs. 3 and 4 with one of the rest points QI or Qa approaching 0 as d -+ 0. The nature of the resulting degenerate rest point at 0 (with d = 0) can be investigated using Theorem 65 of [S] if au + us2f 0 and Theorem 67 of [8] if aI, + us2 := 0. These theorems are stated in the appendix of this paper. The results are just as one would expect: 0 is a saddle-node if a,, + u2?.f 0 and d = 0 and 0 is a degenerate rest point consisting of two hyperbolic sectors if alI + u2s= d -= 0. The separatrix configurations resulting when either Qr or Q3 approaches 0 in the cases of Fig. 3 are shown in Figs. 6(a) and (b), respectively. The separatrix configurations resulting when Q1 approaches 0 in the cases of Fig. 4(a) are shown in Fig. 7(a) if lim,,,(u,, + u,~) # 0 and in Fig. 8 if lim,,,(u,, + uz2) = 0. The separatrix configurations resulting when Qa approaches 0 in the cases of Fig. 4(a) are shown in Fig. 7(b). Similarly, letting Q1 or Q3 approach 0 in Figs. 4(b)-(d) results in the separatrix configurations shown in Figs. 7(c)-(f) or Fig. 8. These results are summarized in the following lemmas. (Note that the hypotheses of Lemma 13 imply that b > 0 and therefore that Q1 = 0 and Qs f 0 for this case.) LEMMA 11. The separutrix configuration for system (7) with j c I < 2, ull < 0, u21 = uz2 = 0 and (i) aI2 + call < 0 is 0-homeomorphic to one of the configurations shown in Fig. 6(a); (ii) aI2 + calI > 0 is 0-homeomorphic to one of the conjigurations shown in Fig. 6(b). LEMMA
12.
The sepuratrix
conjiguration
for
system (7) with
/ c 1 < 2,
all < 0, a21-c 0, ~~~~~~= 42~21 , all + a22f 0 and(i) al2 - a2l + call < 0 is 0-homeomorphic to one of the configurations in Figs. 7(b)-(e); (ii) al2 - a21 + calI > 0 is 0-homeomorphic to one of the configurations in Figs. 7(a) or (f).
266
DICKSON AND PERK0
FIG. 6
(a)
(b)
(d)
(el FIG. 7
LEMMA 13. The separatrix conjiguration for system (7) with 1c 1 < 2, a,, < 0, a,, < 0, a11a22 = a,,a,, and alI + az2 = 0 is 0-homeomorphic to one of the configurations shown in Fig. 8.
BOUNDED
QUADRATIC
SYSTEMS
FIG.
IN
THE
PLANE
267
8
We next determine the possible separatrix configurations in the case of one (finite) rest point at the origin for the bounded cases of system (7); i.e., for 1c 1 < 2 and either (i) a,, = 0, azl f 0, aI2 + u2i = 0 and caZL+ us2 < 0 in which casethe origin is a stable node for u2a < -2 ( ua11,a stable focus for -2 j usi 1 < us2 < min(O, -~a,,) and if cuzl < 0 the origin is a stable focus for uaa = 0 and (by the Poincare-Bendixson Theorem) an unstable focus on the interior of a limit cycle for 0 < u2a< -cup1 , a limit cycle being generated at the origin as u2s becomes positive (It is conjectured that the limit cycle is unique and stable and expands monotonically to a limiting “snail-shaped curve” as ua2 increases to the value -~a,, > 0 (cf. [J], § 6). The limiting “snail curve” is pictured in Fig. 11 below.) (ii) al1 < 0 and b2 < 4d in which case the origin is a stable node or focus for a,, + uaa < 0 or (by the PoincarCBendixson Theorem) an unstable focus in the interior of a limit cycle for a,, + ass > 0, a limit cycle being generated at the origin as ai1 + ua2becomes positive; or (iii) air < 0 and b = d = 0 in which case ai1 + a,, < 0 and it follows from Theorem 65 in [S] that the origin is a stable node. In these cases, it was pointed out in the proof of Lemma 4 that (7) has only one critical point at infinity, a saddle-node as shown in Fig. 1. The above remarks establish the following lemma.
FIG.
9
LEMMA 14. The sepuratrix ronj&uration for system (7) with c .: 2, and either (i) a,, =-: aI2 -I- azl := 0, a,, / 0 and caal i n,, -: 0 or (ii) a,r I. 0 and either b2 C’ 4d or b :m d : : 0 is 0-homeomorphic to OIZE of the conf@rations in Fig. 9.
The global behavior of the bounded cases of system (7) a line of rest points is easily deduced from the related linear azI = 0 or (10) in case aI1 : - a2r t a,, - cazl -i aI1 and the global behavior can be depicted as in Fig. IO if al2
-2LCLO
when there exists system (9) in case a,, -- 0, azl -8 0 == 0 or Fig. 1 1 if
c=o FIG. I1
We next consider the bounded cases of system (3). The bounded cases of (3) are integrable and the solution of (3) in these cases is given in Eq. (4). It is useful in determining the global behavior to note that there are two critical points at infinity, Pr( * 1, 0,O) and P,(O, + 1, 0), and that it follows from Theorem 65 of [S] that Pi is a saddle-node if alI f 0. If az2 = 0 the x,-axis is a line of rest points and the trajectories of (3) coincide with those of the related linear system x1 = al, (11) azl + x2 x2
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
269
for x1 > 0 and with the trajectories of (11) with the direction of motion reversed for x1 < 0. The global behavior in this case is depicted in Fig. 12.
FIG.
12
If ui2 = 0, a,, < 0 and us2 < 0 the local behavior near Pz is not determined by the results of [S]. H owever, it follows from the solution (4) that P, is a saddle-node (this follows since the x,-axis and the equator of S2 consist of trajectories, every trajectory in xi > 0 has the origin as its w-limit set and PI as its a-limit set and every trajectory in xi < 0 has the origin as its w-limit set and P2 as its a-limit set). Also in this case, the origin is a stable node. Thus, we have the following lemma. LEMMA 15. The separatrix conjiguration for system (3) with aI2 = 0, a,, < 0 and az2 < 0 is 0-homeomorphic to the configuration shown in Fig. 13.
FIG.
13
We finally consider the bounded casesof (5). These casesare integrable and the solution is given in Eq. (6). If u2r = us2 = 0 we have a parabola of rest
270
DICKSON L
x1ND
PKRKO
points, xz2 + u12~2+ allq .: 0 and the global behavior in this case is depicted in Fig. 14 for al, < 0.
FIG. 14
If azl = 0, az2 < 0 and aI1 < 0 then the x,-axis consists of trajectories, the origin is a stable node and it follows from (6) that the critical point at infinity P1( k 1, 0, 0) is a saddle-node. We have the following lemma. LEMMA 16. The separatrix configuration for system (5) with aSI = 0, az2 < 0 and al, < 0 is 0-homeomorphic to the configuration in Fig. 9 with -3 interpreted as a stable node.
If uzl = a,, = 0 and a22 < 0 then (5) has x2 = 0 as a line of rest points and the global behavior is determined by the related linear system
The global behavior of (5) in case uzl = a,, = 0 and uz2 < 0 is depicted in Fig. 15.
FIG.
IS
BOUNDED QUADRATIC SYSTEMS IN THE PLANE
271
This completes the determination of all possible phase portraits of bounded quadratic systems in the plane. We summarize these results in the following theorem wherein a phase portrait means an equivalence class of the set of trajectories of a system, two sets of trajectories of a system in E2 being equivalent iff there exists a homeomorphism of E2 carrying trajectories of one onto trajectories of the other in a 1 : 1 manner. THEOREM 2. The phase portrait of a bounded quadratic system is either determined by one of the separatrix conjigurations shown in Figs. 3, 4, 6-9, 13 or it is determined by quadratures as in the cases depicted in Figs. 10-12, 14, 15.
The definition of phase portrait based on equivalence under homeomorphism could just as well be based on equivalence under 0-homeomorphism provided the phase portraits obtained by rotating those of Figs. 4,7,8, and 1I about the x,-axis are included in the glossary of possible types in Theorem 2. This paper then classifies the bounded quadratic system in the plane (Theorem 1) and determines all possible phase portraits for such systems (Theorem 2). It is also a step in the direction of characterizing the phase portraits of all bounded quadratic systems in the plane by means of algebraic inequalities on the coefficients (Lemmas 9-l 6).
APPENDIX
This appendix contains the theorems in [8] that are referred to in this paper concerning the local behavior near critical points of the system % = Ax if(x),
.v E E2
(A.1)
where f (x) is analytic in a neighborhood of the origin and has a power series expansion beginning with second degree terms in x and A is a singular matrix. If det A = 0, tr A f 0 and if the origin is an isolated critical point for (A.l), then (A.1) can be put into the form (cf. [8], p. 372)
1 = p&G Y) 9 = Y t 02(x, Y)
1 (A.2)
where P2 and Q2 are analytic in a neighborhood of the origin and have expansions that begin with second degree terms in x and y. The following theorem covers this case.In that theorem (and in Theorem 67) a critical point, a canonical neighborhood of which consists of one parabolic and two hyperbolic sectors, is called a saddle-node.
272
DICKSON AND PICRKO
THEOREM 65. Let the point (0, 0) be an isolated critical point for the system (A.2). Let y =: p)(x) be the solution of the equation y + 0,(x, y) = 0 in a neighborhood of (0,O) and let the expansion of the function 1,5(x) == Pz(x, F(X)) in powers of x have the form $I(.~) _= A,nxnX -t ... where m >, 2 and A,, -;t- 0. Then (1) for m odd and A,,, > 0 the critical point (0, 0) is a topological node, (2) for m odd and A,,, < 0 the point (0, 0) is a topological saddle and (3) for m even the point (0,O) is a saddle-node. If A # 0 is a singular matrix with tr A = 0 and if the origin is an isolated critical point of (A.l), then (A.]) can he put into the form (cf. [8], p. 379) +$ z.
3'
j
a,#[1
1 (A.3) =
+
h(x)]
-t
b,x~y"y[l
+
g(x)]
-t
y2+,
Y)
where h(x), g(x) and R(x, y) are analytic in a neighborhood of the origin, h(0) = g(0) = 0, K > 2, a, f 0 and n >: 1. The next two theorems treat this case. In these theorems a critical point, a canonical neighborhood of which consists of two hyperbolic sectors, is called a degenerate critical point and a critical point, a canonical neighborhood of which consists of one hyperbolic and one elliptic sector, is called a critical point with an elliptic domain. THEOREM 66.
Let /z
2m t 1 (m -;: I) in (A.3) and let h ~~ b,,” 4 4(m -I- 1) ak .
Then if ak > 0, the critical the critical point (0,O) is:
point (0,O) is topologically
a saddle. If ak < 0,
(1) afocusoracenterifb,=-Oandalsoifb,fOandn>nzorifn=m and X < 0, (2) topologically a node ;f b, # 0, n is an even number and n < m and also zf b, f 0, n is an even number, n = m and A 3 0, (3) a critical point with an elliptic domain afbn f 0, n is an odd number and n < m and also if b, f 0, n is an odd number, n = m and h > 0. THEOREM 67.
Let k = 2m (m > 1) in (A.3). Then the CriticaEpoint
(0,O)
is: (1) a degenerate critical point if b, = 0 and also if b, f
0 and n > m,
(2) a saddle-node if b, # 0 and n < m. The proofs of these theorems are contained in Chapter 9 of [8].
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
273
PLANE
REFERENCES 1. COPPEL, W. A., A survey of quadratic systems, J. Diff. Eq. 2, 293-304 (1966). 2. DUFF, G. F. D., Limit cycles and rotated vector fields, Ann. Math. (2) 57 (1953), 15-31. 3. YEH YEN-CHIEN, A qualitative study of the integral curves of the differential
& Qoo+ QIG + QOIY +Qz& +QSY +Qozy’ I Chinese Mut,, equation d,= PO0 +P,$+ Poty + P2,x2 + P,,xy
4. 5.
6. 7. 8.
9. 20.
+ P,,y”’
’
3 (1963), l-18; II, Uniqueness of limit cycles, Chinese Math. 3 (1963), 62-70. MARKUS, I,., Quadratic differential equations and non-associative algebras, Ann. Math. Studies, No. 4.5, Princeton Univ. Press (1960), 185-213. DICKSON, R. J. AND PERKO, L. M., Quadratic differential systems, Lockheed Research Report No. LMSC/L-56-68-1, Lockheed Palo Alto Research Lab., Feb. 1968. HART~IAN, P., “Ordinary Differential Equations,” Wiley, New York (1964). LEFSCHETZ, S., “Differential Equations: Geometric Theory,” 2nd ed. Interscience, New York (1962). ANDRONOV, A. A., LEONTOVICH, E. A., GORDON, I. I., ?.ND MAIEN, A. L., “The Qualitative Theory of Dynamical Systems of Second Order,” Nauk Publishing House, Moscow, (1966) (in Russian). equations in the plane,” MARKUS, L., “Global structure of ordinary differential Trans. Am. Math. Sot. 76, 127-148 (1954). TUNC, CHIN-CHU, Positions of limit cycles of the system dx - = co
%kX’Y”
4 yg=
&c,+~<~ . .
bnxiyk,
Chinese Math.
3, 227-284
OF DIFFERENTIAL
EQUATIONS
7, 25 l-273
Bounded Quadratic R. J. DICKSON
(1970)
Systems in the Plane* AND L.
Lockheed Palo Alto Research Laboratory, Received November
M.
PERKO
Palo Alto, California
94304
8. 1968
This paper contains a study of those two-dimensional autonomous systems with quadratic polynomial right-hand sides which have all of their trajectories bounded for t > 0. Such systems will be referred to as bounded quadratic systems. It follows from the PoincarbBendixson Theorem that any such system will have a rest point in the plane and by translating the origin to that rest point it will have the form if = Ax -tf&),
XEE2,
(1)
where A is a constant matrix and where the components of fs(~) are homogeneous quadratic polynomials in x = (x1 , x2). It will be assumed that f2(x) + 0 since linear systems can be integrated in terms of elementary functions. The survey paper of Coppel [I] contains most of the important results for quadratic systems in the plane. At the end of his paper, Coppel states that what remains to be done for quadratic systems is to determine all possible phase portraits and, ideally, to characterize them by means of algebraic inequalities on the coefficients. This paper first establishes necessary and sufficient conditions for a two-dimensional quadratic system to have all of its trajectories bounded for t > 0 and then determines all possible phase portraits for such bounded quadratic systems in the plane. In determining all possible phase portraits for bounded quadratic systems, some characterization by means of algebraic inequalities on the coefficients of system (1)-or (1) under a suitable linear transformation of coordinates-is also accomplished. In order to complete the algebraic characterization, there remains the problem of determining algebraic inequalities on the coefficients which decide the number and stability properties of limit cycles around each isolated rest point. This remains the outstanding unsolved problem for bounded quadratic systems in the plane. The limit cycle structure to be found in bounded plane systems * This work was supported by the Air Force Office of Scientific Research under contract AF 49(638)-1685 and by the Lockheed Independent Research Program. 251 505/7/z-4
252
DICKSON AND PERK0
can be investigated by using results known for general quadratic systems, (cf. [I], pp. 295-298) and by applying the theory of rotated vector fields [2]. This has been carried out for certain (unbounded) quadratic systems in the plane by Yeh Yen-Chien [3] together with an investigation of the uniqueness of limit cycles. While this approach is quite promising, our results are not not sufficiently complete to justify their exposition here.
1.
CLASSIFICATION
OF ROUNDED QUADRATIC
SYSTEMS
Markus has classified the homogeneous quadratic systems in the plane up to affine transformation by classifying the related real linear algebras; (cf. Theorems 6-8 of [Kj). Th e h omogeneous quadratic systems corresponding to the algebras in Theorems 7 and 8 and to those in Theorem 6, Cases (3), (4), (6), (8), and (10) with k y) -& all have ray solutions; i.e., the related algebra has an idempotent element. Now if the homogeneous quadratic system
has a ray solution then the quadratic system (1) will have an unbounded solution; (cf. Theorem 2, $3 in [.5]). Th us, if fi(x) f 0, the system (1) has an unbounded solution unless the quadratic part of the right-hand side is linearly equivalent to one of the following forms (corresponding to the algebras of Theorem 6, Cases (2), (5), (7), (9), and (10) for k < -4, the last two cases corresponding to the last form below):
0. iXl% 11 .X22 i 0 1;
(A) w (Cl
i
03
( -x1x2
Xl%! -I- x22 1; q2
X22 + L-x221 ’
ICI
(2.
First let us show that the system (1) withf,(x) given in (C) has an unbounded trajectory (even though the corresponding homogeneous quadratic system has no ray solution). LEMMA
1.
The system 9 = Ax
+ (
%X2 +x22 , > x22
x(0)
= xg
has an unbounded trajectory (as t -+ co) for somex0 E E2.
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
253
PLANE
Proof. Let D,, = {(x 1 , x2) : xl 3 01,x2 > /I} where 01,fl > 0. Since the only case when the rest points of (2) are not finite in number is when xs = --al, is a line of rest points (i.e., when us1 = 0 and either a,, = 0 or a,, = ass), it follows that for any given matrix A, 01and p can be chosen sufficiently large that D,, contains no rest points of (2). On x1 = CL,
which is positive for all 01> 0 and xs > i3 provided /I is sufficiently large. On x2 = 8, 3i2
=
+ w
a2121
+
a22h
which for u21 > 0 is positive for all /3 > 0 and x1 2 01provided 01is sufficiently large. For u21 = 0 the above expression for 3i’,is positive for ,8 > 0 sufficiently large. Thus, for usI > 0 and the parameters 01and /I sufficiently large, Dae is a positively invariant set which contains no rest points of (2) and therefore the system (2) has an unbounded trajectory (as t 4 00) for some x,, E D,, . This follows, for example, from Hartman’s Theorem 1.1, (p. 202 in [6]). For u21 < 0 consider the curve V(xl , x2) ZG -x1 + kzz2 = 0 for x2 > 0. The flow defined by (2) is upward across V(X, , x2) = 0 for 0 < k < l/(2 ) u2r I) and x2 sufficiently Iarge. This follows since from (2) and the definition of Wl Px2) dV = -2, dt
+ 2kx2d2 = k[l + 2a,,k] x23 + 0(x,2)
> 0
a21I) and x2 sufficiently large. Let
onV(x,,x,)=OforO
D,, = {(x1 7x2) : x1 >, 01,x2 > (xJk)“2}. Then for a,,
2.
The system
ji=klx+
0
( 1)
x(0) E z?
XIX2
has all of its trajectories bounded (for t > 0) if and only if uI2 = 0, a,, < 0 and a22 < 0. Proof. The critical points at infinity for (3) can be represented by the pairs of diametrically opposite points PI( &l, 0,O) and Pz(O, &l, 0) on the
254
DICKSON
AND
PERK0
equator of the unit sphere in E3 (cf. e.g., Lefschetz [?‘I, p. 201). In terms of local coordinates (x1 , x8) at the point P, , Eq. (3) has the form
For ula f 0 this system has an elliptic sector at (0, 0). This follows, for example, from Theorem 66 on p. 297 of Andronov and Leontovich [S], which is stated in the appendix of this paper. This implies the existence of unbounded trajectories for (3) as t + co in cast a,, + 0. For ai2 =: 0, Eq. (3) is integrable. We obtain for ata _= 0, a,, :z 0 \
x1(t) = Xl(O) &It x2(t) = x,(O) exp [a&
:- F
+‘I
+ xl(O) azl ‘t exp a,,(t I ‘01
Xl(O) - t’) f a,,t’ f- __ (e”“t _ e”“t’ )]df’ alI 1. (4)
If a,, > 0 and ~~(0) $0 then j sr(t)i - w as t--f co. If a,, < 0, xl(t) as t-p COand it follows in this case that / x2(t)l -
M h I x2(O)/ + I x,(O) a,,/a,,
0
i
as t + Co if a 22 < 0 and that 1q(t)! --f co if a2a > 0, X%(O)f 0 and ~~(0) = 0. The remaining case when ur2 = all = 0 is integrable and obviously has unbounded solutions (as t---f co) for ~~(0) sufficiently large. This completes the proof of Lemma 2. Let us next consider the system (1) withf,(x) given in (B). LEMMA
3.
The system
*==A+ EE2 c-Q2 01’ x(0)
(5)
has all of its trajectories bounded (for t > 0) if and only if uzl = 0, a,, .< 0, u22 ,( 0 and alI + az2 < 0. Proof. The system (5) by the pair of diametrically unit sphere in Es. In terms form 32, = -a21x3 2, =
x3(a$3
has only one critical point at infinity represented opposite points Pr( & 1, 0, 0) on the equator of the of local coordinates (xZ , x3) at Pr , Eq. (5) has the + (all -- ad x2x3 + +23222x3+ xa3 +
%$Zx3
+
%“)*
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
255
PLANE
For a2r f 0 this system has a node at (0,O). This follows, for example, from Theorem 66 of [8]. The existence of a node at infinity for a system of even parity such as (5) implies the existence of unbounded trajectories (as t + UJ). For u2i = 0, Eq. (5) is integrable. We find that for a,, = 0
xl(t) = xl(0)enllt
alzxz(o) [exp[(apz - a,&] - I] if a,, f
+ eallt
u22 -
aI,
uz2
if a,, -= uzgI/
/ %2%(O) t ’ x2’(O) + eallt 2u,, - a,, [expP22 r %2(o) t
- u,Jt] - I] if a,, f
2u,, ‘1 ’
if a11 == 2a9,, -‘. !I
I ” (6)
x2(t) = x2(O) eazzt
Thus, when usi = 0, it follows that if a,, < 0 and u22 < 0 then xi(t) and x,(t)--tOast--tco;ifu,,>Oandx,(O)fOthenjx,(t)(~cciast--,co; if a,, > 0, x2(O) = 0 and x,(O) # 0 then / xl(t)\ - co as t - co; if al1 -z 0, us2 < 0 then x2(t) + 0 and xl(t) --f x,(O) - u,,~,(O)/u~~ - .zs2(0)/2a,, ; if U 22 = 0, a,, < 0 then x2(t) = x2(O) and xl(t) +- -~~~x~(O)/u,r -- x,2(0)/u,, ; and finally, if u,r = uz2 = 0 and xz(0)(x2(O) + czr2)f 0 then 1x,(t)\ + co as t + co. This completes the proof of Lemma 3. Lastly, we consider the system (1) with &(x) given in (D). LEMMA
4.
The system Le=llAx+
X22 i -x1x2
+ cx2 2'i
x(0) E E2
(7)
has all of its trajectories bounded (for t > 0) if and only if j c 1 < 2 and one of the following sets of conditions: (i) a,, < 0; (ii) a,, = 0 and u21 = 0; OY (iii) a,, = 0, %l# 09 a12 + a21 = 0 and cu,, + u22 < 0 is sutis$ed. Proof. If 1c / > 2, the homogeneous quadratic system corresponding to (7) has a ray solution (Theorem 6 in [d]) and it follows from Theorem 2, $3 in [5] that (7) has an unbounded solution. If ( c / < 2 then (7) has only one critical point at infinity represented by the pair of diametrically opposite points P,(fl, 0,O) on the unit sphere in l?. In terms of local coordinates (x2 ) xa) at PI , Eq. (7) has the form
2, = --x2 + u&c3 - (Ull - uz2) x2x3 + cx22 - u12x2%3- x23 . 1 3i’, = -x3(ullx3 + u12X2X3 +x22)
(8)
It follows from Bendixson’s theorem, (cf. Lefschetz [7], p. 24), that the above system (8) has a node, a saddle or a saddle-node (two hyperbolic sectors and a fan) at (0, 0), according to whether the index at (0,O) is respectively
256
DIC’KSON
AND
PERK0
+I, -1, or 0. Theorem 65 in Andronov and Leontovich [8], p. 379, which is stated in the appendix of this paper, is used to obtain the following results: If all f 0, then (8) has a saddle-node at (0, 0); if a,, =- 0 and a,,(ar, + CZ& > 0, then (8) has a node at (0,O); if a,, = 0 and ~,,(a~, + a,,) < 0 then (8) has a saddle at (0,O); if a,, = 0, ula + a,, - 0 and u,,(cu,, + uaa)f 0, then (8) has a saddle-node at (0,O). The existence of a node or saddle at infinity for a system of even parity such as (7) implies the existence of an unbounded trajectory (as t + co). Hence, if a,, = 0 and a,,(~,, + a,J f 0, (7) has an unbounded trajectory (as t--t CD). The behavior of (7) for / .x / > 1; i.e., near the equator of the PoincarC sphere as determined by (8) is shown in Fig. 1 for a,, < 0 and also for the case ai1 = 0, ui2 + u2i = 0 and a~,(caal + uaa) < 0; it is shown in Fig. 2 for
FIG.
1
FIG. 2 all > 0 and also for the case a,, = 0, ura + u2r = 0 and u~,(cu,, + uz2) > 0. It follows from the local behavior of (7) near the critical point PI at infinity that all trajectories of (7) are bounded (for t 3 0) if a,, < 0 or if urr = 0, a,, + us1 = 0 and &(~a,, + u2a) < 0 and that (7) has unbounded trajectories (ast-+co)ifu,,>Oorifu,, =0, aI2 + uzl = 0 and a~i(ca,r + u2J > 0. If a,, = us1 = 0 then (7) has x2 as a common factor (i.e., x2 = 0 is a line
BOUNDED QUADRATIC SYSTEMS IN THE PLANE
257
of rest points) and the global behavior of (7) follows from the related linear system $1 =
012 +x2
R, = a22 -
(9)
I*
x1+cx2
This system has a center or a focus at (aZ2- ca12, -al,) according to whether c is equal to zero or not. It follows that with a,, = azl = 0 and / x,, 1 sufficiently large, all trajectories of (7) approach the line of rest points x2 = 0 as t---f co. If a,, = 0, azl f 0 and aI2 + aZ1= caal + a22 = 0 then (7) has (x2 + a,,) as a common factor (i.e., x2 = -alp is a line of rest points) and the global behavior of (7) follows from the related linear system
This system has a focus or center at (0,O) according to whether c is equal to zero or not. It follows that with al, = 0, azl f 0, a,, + uzl = capI + az2 == 0 and / x,, 1 sufficiently large, all trajectories of (7) approach the line of rest points x2 = -aI2 as t -+ co. This completes the proof of Lemma 4. The above results are summarized in the following theorem. THEOREM 1. The quadratic system (1) (with f2(x) + 0) has al2 of its trajectories bounded for t > 0 if and only tf there exists a linear transformation which reduces it to one of the systems (3), (5), or (7) satisfying the conditions of Lemmas 2, 3, or 4, respectively.
2.
CENTERS FOR BOUNDED
QUADRATIC
SYSTEMS
Necessary and sufficient conditions for the existence of a center for a quadratic system in the plane are given in the survey paper of Coppel [I], p. 295. These results are used to determine necessary and sufficient conditions for the existence of a center for systems (3), (5), and (7). A system of the form (1) has a center at (0,O) only if the origin is a center for the corresponding linear system; i.e., only if a11a22- a,,~,, > 0 and all + a22 = 0. This implies that apI # 0 and therefore
is a nonsingular matrix with the property that B-lAB
= [_ol
;I.
258
DICKSON AND PERK0
With x = By, Eq. (1) becomes 9 = B-lABy
+ B-$(By)
which is equivalent to Eq. (3), p. 295 in [I]. The algebraic conditions of [I], p. 295 which are necessary and sufficient for the above system to have a center at (0,O) then yield the following results for the systems (3), (5), and (7), respectively: LEMMA 5.
The system (3) has a center at (0,O) if and only ;f a11 = as” = 0
LEMMA 6.
a@12 < 0.
The system(5) has a center at (0,O) if and onb if w22
LEMMA 7.
and
- a12a2, > 0
and
aI1 +
~2~
= 0.
The system (7) has a center at (0, 0) ;f and only ;f c = a,, = az2 = 0
and
a12az1< 0.
The above results will be used in the next section devoted to determining all possible phase-portraits for bounded quadratic systems.
3. PHASE-PORTRAITS FOR BOUNDED QUADRATIC SYSTEMS Markus [9] has shown that in the plane two Cl systems with isolated critical points and no limit separatrices are equivalent if and only if their separatrix configurations are equivalent. Thus, in case (1) has only a finite number of rest points, it suffices to determine all possible separatrix configurations for (I) in order to determine all possible phase portraits for (1). All possible separatrix configurations for the bounded cases of (1) with only a finite number of rest oints are determined in this section. On the other hand, if (1) does not have a %nite number of rest points, it has a line or curve of rest points and the global behavior is determined by a related linear system. In this way, all possible phase portraits for bounded quadratic systems in the plane are determined in this section. Tung Chin-chu [IO] showed that each limit cycle of (1) contains exactly one rest point in its interior (cf. Theorem 2, p. 296 in [I]). The number of limit cycles around each isolated rest point of (3), (5), or (7) will not be taken up in this paper, since only partial results are available at this time. It is therefore necessary to use the symbols --t-O and 4 in order to exhibit all possible phase-portraits for bounded quadratic systems in the plane. The
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
259
first symbol denotes either a stable node or focus, or a stable or unstable focus on the interior of one or more limit cycles, the outermost of which is externally stable. The second symbol above is similarly defined, the words stable and unstable being interchanged in making the definition. Theorem 6 on p. 299 in [I] which states that a critical point on the interior of a closed path must be either a focus or a center is implicitly used in deriving the results in this section. Let us first consider the system (7) under the conditions of Lemma 4. This is the most interesting case of a bounded quadratic system since the bounded cases of (3) and (5) are integrable. The critical point at infinity P,(&l, 0, 0) for system (7) with j c j < 2 has been discussed in the proof of Lemma 4. Regarding the finite critical points of (7), we have the following lemma. Let d = u1ra2a- u12u21and b = aI3 - a,, + cu,, . LEMMA 8. If a,, f 0, then (7) h as: thyee (jinite) rest points i# d f 0 and b2 > 4d; two (finz’te) rest points ajf b f 0 and either b2 = 4d OY d = 0; one (finite) rest point 23 either b2 < 4d or b = d = 0. In the cuse of three (Jinite) rest points, Qi(xli, xzi) i = 1, 2, 3, it follows that (with proper indexing): x21 < xz2 < x,~; Q2 is a saddle, and Ql and Q3 are either nodes oy foci; Q3 lies to the right (kft) of the line QlQ2 ifull < 0 (a,, > 0).
Proof.
If a,, f 0 then Ax+(
-x,x,
x22 )=o + cx2”
has the solutions x1 = x2 = 0 and x1* = -
(a12 + X2*) x2* %I
with x2* given by x2
* _ -b -
i
(b2 - 4d)1/2
2
The results on the number of (finite) rest points follow immediately. It also follows from the above equations for the rest points that x2- < 0 < x2+ iff d < 0, i.e., iff the origin is a saddle; that x2- < x2+ < 0 iff d > 0 and b > 0; and that 0 < x2- < x2+ iff d > 0 and b < 0. This implies the results stated for the case of three rest points (d f 0, b2 > 4d): First that the x,-coordinates of the rest points are ordered and then (by translating the origin to each of the three rest points, a transformation which leaves the form of the equation as well as the coefficient a,, invariant) that the middle rest point (in the x,-sense) must be a saddle and the upper and lower rest points
260
DICKSON
AND
PERKO
must be nodes or foci. This follows since the above results imply that the coefficient matrix of the linear terms at the upper and lower rest points must have a positive determinant and since these rest points cannot be centers for (7) with aI, f 0 according to Lemma 7. --The fact that ,O+ lies to the right of the line -_lrOO- when a,, < 0 follows from the fact that with the saddle at 0 the line Q+Q- intersects the positive X, axis; i.e., the equation of the line Q+,O- ,
with x2 = 0 yields (with a,, < 0 and d < 0) Xl =
x2+x1x2
-- x2-x1+ + -
x2-
> 0.
This follows since x2+ > x2- and since the above equations for x1* imply that x2+x1- - x2-x1+ > 0 for aI, < 0 and d < 0. (The results for aI, > 0 are proved in an analogous manner.) This completes the proof of Lemma 8. Regarding the finite critical points of (7) in the bounded cases of (7) not covered in Lemma 8, we have one critical point at the origin if 0 and cazl + az8 < 0 and a line of rest points if a11 = 62 + 41 = 0, uzl f a,, = u21 = 0 or if aI, = aI, -+ uzl -= cuzl + uz2 = 0. We now determine all possible separatrix configurations in the case of three finite rest points for the bounded casesof system (7); i.e., for / c 1 < 2, a,, < 0, d f 0 and b2 > 4d. It is no restriction to assume that the saddle is at the origin since translating the origin to any rest point of (7) leaves the form of the equation as well as alI invariant; i.e., we assume that d < 0. Then it is no restriction to further assume that a21< 0 since the transformation x1 -9 x1 , x2 + -x2 transforms (7) into
R, = --a,,~,
+ az2x2 - xlxz - cxz2
which leaves the form of Eq. (7) as well as a,, and d invariant. Note, however, that this transformation is not orientation preserving. In stating the following lemmas, we use the term 0-homeomorphism for an orientation preserving homeomorphism. LEMMA
9.
The separatrix
con$gz/ration fog the system (7) with ) c / < 2,
a,, < 0, azl = 0, and d < 0 is 0-homeomorphic to one of the con$gurations shown in Fig. 3.
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
261
FIG. 3 Proof. It was shown in the proof of Lemma 4 that for 1c 1 < 2 and a,, < 0 the only critical point at infinity is the saddle-node at P,(f 1, 0,O). The behavior near infinity, i.e., near the equator of the PoincarC sphere, as determined by Eq. (8) is shown in Fig, 1 for this case. Note that the equator consists of separatrices. If us1 = 0 and a,, < 0 the xi-axis is composed of trajectories with k1 < 0 for xi > 0 and f, > 0 for x1 < 0. If d < 0, the x,-axis consists of three separatrices (including the origin) with (0,O) as their w-limit set. If d < 0 there are two separatrices (besides the origin) with the saddle at the origin as their a-limit set, one in the upper half plane, Ti , and one in the lower half plane, T, . If d < 0 there is one rest point in the upper half plane and one in the lower half plane (Lemma 8). The PoincarCBendixson Theorem then implies that either the rest point in the upper half plane or an externally-stable limit cycle around that rest point is the w-limit set of Tr and that the rest point in the lower half plane or an externally-stable limit cycle around that rest point is the w-limit set of T, . This completes the proof of Lemma 9, the only possible separatrix configurations for this case being 0-homeomorphic to one of those in Fig. 3.
LEMMA 10.
The separatrix conjiguration for the system (7) with ( c [ < 2, a,, < 0, a,, < 0 and d < 0 is 0-homeomorphic to one of the configurations shown in Fig. 4. Proof. It follows from Lemma 8 that under the given hypothesis, (7) has three (finite) rest points; one at the origin 0, one in the upper half plane Q1 and one in the lower half plane Qs to the right of the line OQ1 . According to the result of Tung Chin-chu given on p. - 296 of - [I], the flow on the line @, is in the same sense on the segments co0 and Qlco and in the opposite sense on the segment OQ1 . Similar results follow for the flow on 0% (cf. Fig. 5). The flow on the x,-axis for u2i < 0 is clockwise (except at x1 = 0).
262
DICKSON
ASD
(a)
PERK0
(b)
(d) FIG. 4
FIG. 5
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
263
PLANE
_The x,-axis and the lines OQ1, OQa divide s” into six open sectors Ri , i = l,..., 6 numbered with increasing argument, R, being in the upper half plane and having the positive x,-axis as part of its boundary. The flow on the x,-axis and on the lines OQ1 and OQa near the origin establishes the fact that (with d < 0) the two separatrices which have the saddle at the origin as their w-limit set, T1 and T, , approach (0,O) as t + co in the sectors R, and R, , respectively; and that the two separatrices which have (0,O) as their a-limit set, T3 and T4, approach (0,O) as t - --cc, in the sectors R, and R, , respectively. It follows from Eq. (8) d escribing the behavior near P, that for aa < 0 the separatrix T5 with PI as its a-limit set approaches PI as t -+ -CC in sector R, (cf. Fig. 5). Now since the boundary of R, as a subset of S2 consists of points of egress (the lines OP, and OW), a trajectory (on the equator), a saddle at 0 and a saddle-node PI , it follows from the Poincart-Bendixson Theorem that the saddle-node PI must be the a-limit set of T, as is shown in Fig. 5. Similarly, considering the boundary of R5 v R, , it follows that Qa or an externally stable limit cycle around Qa must be the w-limit set of the separatrices T4 and T, as is shown in Fig. 5. Finally, considering the boundary of R, V R, u R, u R, , it follows from the Poincare-Bendixson Theorem that the separatrix T3 must have one of the following sets for its w-limit set: (i) the rest point Qr or an externally-stable limit-cycle about Q1 , in which case Tl has Pl( - 1, 0,O) as its a-limit set, as in Fig. 4(a), (ii) the saddle at (0,O); i.e., T, coincides with T, and is a separatrixcycle which contains the rest point Q1 (cf. Theorem 2, p. 296 in [I]) as well as any limit cycle around Q1 , as in Figs. 4(b) and (c) or (iii) the rest point Qa or an externally-stable limit-cycle around Qs , in which case Ti has Qi or an externally-unstable limit-cycle around Q1 as its a-limit set, as in Fig. 4(d). This completes the proof of Lemma 10. Let 7 be the trace of the coefficient to the upper rest point (xi+, ~a+).
matrix
with
the origin
translated
(It is easily shown that 7 = a,, + ua2 + 2cx,+ - xi+.) Regarding the variation of the separatrix configuration in Lemma 10 with respect to a variation of the parameter 7, it is conjectured that we have one of two possible types of behavior, depending on the path in the parameter space (a11 > $2 7 a217 a22 9 c) that corresponds to the variation of 7.
Conjecture. As the parameter 7 increases continuously negative to a large positive value, the separatrix configuration
from a large for the system
264
DICKSON AND PERK0
(7) satisfying the hypotheses of Lemma 10 undergoes one of two possible continuous deformations starting with the configuration 4(a), the upper rest point being a stable node (with no limit-cycle around it): either (i) we have the configuration 4(a) for 7 6: 0 with no limit cycle around the upper rest point, the upper rest point changing from a stable node to a stable focus as T increases through negative values; as T becomes positive, a stable limit-cycle is generated at the upper rest point in 4(a), containing the upper rest point as an unstable focus; as T increases, the limit cycle expands until it intersects the saddle point at the origin and becomes a separatrix cycle at some critical value T* > 0 and we have the configuration 4(c) with no limit cycle around the unstable focus at the upper rest point; finally, as 7 increases through values greater than T*, we have the configuration in 4(d) with no limit cycle around the upper rest point, the upper rest point being first an unstable focus and then becoming an unstable node as T increases without bound; or (ii) we have the configuration 4(a) for 7 less than some critical T* < 0 at which value the configuration 4(b) occurs, the upper rest point going from a stable node to a stable focus as r increases through negative values T << T* and there is no limit cycle around the upper rest point for T .<
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
265
rest point (including the case with one cycle around each of these rest points) and also examples of the two possible variations of the separatrix configuration with the parameter 7 described in the above conjecture have been obtained on an analog computer. We next determine all possible separatrix configurations in the case of two (finite) rest points for the bounded cases of system (7); i.e., for 1c [ < 2, a,, < 0, b f 0 and either d = 0 or b2 = 4d. We may assume that d = 0, for if d f 0 and b2 = 4d then translating the origin to the second rest point yields a system of the same form as (7) with aI, < 0 left invariant and with the determinant of the transformed matrix A equal to zero. We also note that d = 0 and / c j < 2 imply that b f 0. The same type of argument used in Lemma IO can be employed in this case; however, it is more straightforward to observe that due to the continuity of solutions with respect to the parameter d, letting d + 0 through negative values will result in a continuous deformation of the separatrix configurations in Figs. 3 and 4 with one of the rest points QI or Qa approaching 0 as d -+ 0. The nature of the resulting degenerate rest point at 0 (with d = 0) can be investigated using Theorem 65 of [S] if au + us2f 0 and Theorem 67 of [8] if aI, + us2 := 0. These theorems are stated in the appendix of this paper. The results are just as one would expect: 0 is a saddle-node if a,, + u2?.f 0 and d = 0 and 0 is a degenerate rest point consisting of two hyperbolic sectors if alI + u2s= d -= 0. The separatrix configurations resulting when either Qr or Q3 approaches 0 in the cases of Fig. 3 are shown in Figs. 6(a) and (b), respectively. The separatrix configurations resulting when Q1 approaches 0 in the cases of Fig. 4(a) are shown in Fig. 7(a) if lim,,,(u,, + u,~) # 0 and in Fig. 8 if lim,,,(u,, + uz2) = 0. The separatrix configurations resulting when Qa approaches 0 in the cases of Fig. 4(a) are shown in Fig. 7(b). Similarly, letting Q1 or Q3 approach 0 in Figs. 4(b)-(d) results in the separatrix configurations shown in Figs. 7(c)-(f) or Fig. 8. These results are summarized in the following lemmas. (Note that the hypotheses of Lemma 13 imply that b > 0 and therefore that Q1 = 0 and Qs f 0 for this case.) LEMMA 11. The separutrix configuration for system (7) with j c I < 2, ull < 0, u21 = uz2 = 0 and (i) aI2 + call < 0 is 0-homeomorphic to one of the configurations shown in Fig. 6(a); (ii) aI2 + calI > 0 is 0-homeomorphic to one of the conjigurations shown in Fig. 6(b). LEMMA
12.
The sepuratrix
conjiguration
for
system (7) with
/ c 1 < 2,
all < 0, a21-c 0, ~~~~~~= 42~21 , all + a22f 0 and(i) al2 - a2l + call < 0 is 0-homeomorphic to one of the configurations in Figs. 7(b)-(e); (ii) al2 - a21 + calI > 0 is 0-homeomorphic to one of the configurations in Figs. 7(a) or (f).
266
DICKSON AND PERK0
FIG. 6
(a)
(b)
(d)
(el FIG. 7
LEMMA 13. The separatrix conjiguration for system (7) with 1c 1 < 2, a,, < 0, a,, < 0, a11a22 = a,,a,, and alI + az2 = 0 is 0-homeomorphic to one of the configurations shown in Fig. 8.
BOUNDED
QUADRATIC
SYSTEMS
FIG.
IN
THE
PLANE
267
8
We next determine the possible separatrix configurations in the case of one (finite) rest point at the origin for the bounded cases of system (7); i.e., for 1c 1 < 2 and either (i) a,, = 0, azl f 0, aI2 + u2i = 0 and caZL+ us2 < 0 in which casethe origin is a stable node for u2a < -2 ( ua11,a stable focus for -2 j usi 1 < us2 < min(O, -~a,,) and if cuzl < 0 the origin is a stable focus for uaa = 0 and (by the Poincare-Bendixson Theorem) an unstable focus on the interior of a limit cycle for 0 < u2a< -cup1 , a limit cycle being generated at the origin as u2s becomes positive (It is conjectured that the limit cycle is unique and stable and expands monotonically to a limiting “snail-shaped curve” as ua2 increases to the value -~a,, > 0 (cf. [J], § 6). The limiting “snail curve” is pictured in Fig. 11 below.) (ii) al1 < 0 and b2 < 4d in which case the origin is a stable node or focus for a,, + uaa < 0 or (by the PoincarCBendixson Theorem) an unstable focus in the interior of a limit cycle for a,, + ass > 0, a limit cycle being generated at the origin as ai1 + ua2becomes positive; or (iii) air < 0 and b = d = 0 in which case ai1 + a,, < 0 and it follows from Theorem 65 in [S] that the origin is a stable node. In these cases, it was pointed out in the proof of Lemma 4 that (7) has only one critical point at infinity, a saddle-node as shown in Fig. 1. The above remarks establish the following lemma.
FIG.
9
LEMMA 14. The sepuratrix ronj&uration for system (7) with c .: 2, and either (i) a,, =-: aI2 -I- azl := 0, a,, / 0 and caal i n,, -: 0 or (ii) a,r I. 0 and either b2 C’ 4d or b :m d : : 0 is 0-homeomorphic to OIZE of the conf@rations in Fig. 9.
The global behavior of the bounded cases of system (7) a line of rest points is easily deduced from the related linear azI = 0 or (10) in case aI1 : - a2r t a,, - cazl -i aI1 and the global behavior can be depicted as in Fig. IO if al2
-2LCLO
when there exists system (9) in case a,, -- 0, azl -8 0 == 0 or Fig. 1 1 if
c=o FIG. I1
We next consider the bounded cases of system (3). The bounded cases of (3) are integrable and the solution of (3) in these cases is given in Eq. (4). It is useful in determining the global behavior to note that there are two critical points at infinity, Pr( * 1, 0,O) and P,(O, + 1, 0), and that it follows from Theorem 65 of [S] that Pi is a saddle-node if alI f 0. If az2 = 0 the x,-axis is a line of rest points and the trajectories of (3) coincide with those of the related linear system x1 = al, (11) azl + x2 x2
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
PLANE
269
for x1 > 0 and with the trajectories of (11) with the direction of motion reversed for x1 < 0. The global behavior in this case is depicted in Fig. 12.
FIG.
12
If ui2 = 0, a,, < 0 and us2 < 0 the local behavior near Pz is not determined by the results of [S]. H owever, it follows from the solution (4) that P, is a saddle-node (this follows since the x,-axis and the equator of S2 consist of trajectories, every trajectory in xi > 0 has the origin as its w-limit set and PI as its a-limit set and every trajectory in xi < 0 has the origin as its w-limit set and P2 as its a-limit set). Also in this case, the origin is a stable node. Thus, we have the following lemma. LEMMA 15. The separatrix conjiguration for system (3) with aI2 = 0, a,, < 0 and az2 < 0 is 0-homeomorphic to the configuration shown in Fig. 13.
FIG.
13
We finally consider the bounded casesof (5). These casesare integrable and the solution is given in Eq. (6). If u2r = us2 = 0 we have a parabola of rest
270
DICKSON L
x1ND
PKRKO
points, xz2 + u12~2+ allq .: 0 and the global behavior in this case is depicted in Fig. 14 for al, < 0.
FIG. 14
If azl = 0, az2 < 0 and aI1 < 0 then the x,-axis consists of trajectories, the origin is a stable node and it follows from (6) that the critical point at infinity P1( k 1, 0, 0) is a saddle-node. We have the following lemma. LEMMA 16. The separatrix configuration for system (5) with aSI = 0, az2 < 0 and al, < 0 is 0-homeomorphic to the configuration in Fig. 9 with -3 interpreted as a stable node.
If uzl = a,, = 0 and a22 < 0 then (5) has x2 = 0 as a line of rest points and the global behavior is determined by the related linear system
The global behavior of (5) in case uzl = a,, = 0 and uz2 < 0 is depicted in Fig. 15.
FIG.
IS
BOUNDED QUADRATIC SYSTEMS IN THE PLANE
271
This completes the determination of all possible phase portraits of bounded quadratic systems in the plane. We summarize these results in the following theorem wherein a phase portrait means an equivalence class of the set of trajectories of a system, two sets of trajectories of a system in E2 being equivalent iff there exists a homeomorphism of E2 carrying trajectories of one onto trajectories of the other in a 1 : 1 manner. THEOREM 2. The phase portrait of a bounded quadratic system is either determined by one of the separatrix conjigurations shown in Figs. 3, 4, 6-9, 13 or it is determined by quadratures as in the cases depicted in Figs. 10-12, 14, 15.
The definition of phase portrait based on equivalence under homeomorphism could just as well be based on equivalence under 0-homeomorphism provided the phase portraits obtained by rotating those of Figs. 4,7,8, and 1I about the x,-axis are included in the glossary of possible types in Theorem 2. This paper then classifies the bounded quadratic system in the plane (Theorem 1) and determines all possible phase portraits for such systems (Theorem 2). It is also a step in the direction of characterizing the phase portraits of all bounded quadratic systems in the plane by means of algebraic inequalities on the coefficients (Lemmas 9-l 6).
APPENDIX
This appendix contains the theorems in [8] that are referred to in this paper concerning the local behavior near critical points of the system % = Ax if(x),
.v E E2
(A.1)
where f (x) is analytic in a neighborhood of the origin and has a power series expansion beginning with second degree terms in x and A is a singular matrix. If det A = 0, tr A f 0 and if the origin is an isolated critical point for (A.l), then (A.1) can be put into the form (cf. [8], p. 372)
1 = p&G Y) 9 = Y t 02(x, Y)
1 (A.2)
where P2 and Q2 are analytic in a neighborhood of the origin and have expansions that begin with second degree terms in x and y. The following theorem covers this case.In that theorem (and in Theorem 67) a critical point, a canonical neighborhood of which consists of one parabolic and two hyperbolic sectors, is called a saddle-node.
272
DICKSON AND PICRKO
THEOREM 65. Let the point (0, 0) be an isolated critical point for the system (A.2). Let y =: p)(x) be the solution of the equation y + 0,(x, y) = 0 in a neighborhood of (0,O) and let the expansion of the function 1,5(x) == Pz(x, F(X)) in powers of x have the form $I(.~) _= A,nxnX -t ... where m >, 2 and A,, -;t- 0. Then (1) for m odd and A,,, > 0 the critical point (0, 0) is a topological node, (2) for m odd and A,,, < 0 the point (0, 0) is a topological saddle and (3) for m even the point (0,O) is a saddle-node. If A # 0 is a singular matrix with tr A = 0 and if the origin is an isolated critical point of (A.l), then (A.]) can he put into the form (cf. [8], p. 379) +$ z.
3'
j
a,#[1
1 (A.3) =
+
h(x)]
-t
b,x~y"y[l
+
g(x)]
-t
y2+,
Y)
where h(x), g(x) and R(x, y) are analytic in a neighborhood of the origin, h(0) = g(0) = 0, K > 2, a, f 0 and n >: 1. The next two theorems treat this case. In these theorems a critical point, a canonical neighborhood of which consists of two hyperbolic sectors, is called a degenerate critical point and a critical point, a canonical neighborhood of which consists of one hyperbolic and one elliptic sector, is called a critical point with an elliptic domain. THEOREM 66.
Let /z
2m t 1 (m -;: I) in (A.3) and let h ~~ b,,” 4 4(m -I- 1) ak .
Then if ak > 0, the critical the critical point (0,O) is:
point (0,O) is topologically
a saddle. If ak < 0,
(1) afocusoracenterifb,=-Oandalsoifb,fOandn>nzorifn=m and X < 0, (2) topologically a node ;f b, # 0, n is an even number and n < m and also zf b, f 0, n is an even number, n = m and A 3 0, (3) a critical point with an elliptic domain afbn f 0, n is an odd number and n < m and also if b, f 0, n is an odd number, n = m and h > 0. THEOREM 67.
Let k = 2m (m > 1) in (A.3). Then the CriticaEpoint
(0,O)
is: (1) a degenerate critical point if b, = 0 and also if b, f
0 and n > m,
(2) a saddle-node if b, # 0 and n < m. The proofs of these theorems are contained in Chapter 9 of [8].
BOUNDED
QUADRATIC
SYSTEMS
IN
THE
273
PLANE
REFERENCES 1. COPPEL, W. A., A survey of quadratic systems, J. Diff. Eq. 2, 293-304 (1966). 2. DUFF, G. F. D., Limit cycles and rotated vector fields, Ann. Math. (2) 57 (1953), 15-31. 3. YEH YEN-CHIEN, A qualitative study of the integral curves of the differential
& Qoo+ QIG + QOIY +Qz& +QSY +Qozy’ I Chinese Mut,, equation d,= PO0 +P,$+ Poty + P2,x2 + P,,xy
4. 5.
6. 7. 8.
9. 20.
+ P,,y”’
’
3 (1963), l-18; II, Uniqueness of limit cycles, Chinese Math. 3 (1963), 62-70. MARKUS, I,., Quadratic differential equations and non-associative algebras, Ann. Math. Studies, No. 4.5, Princeton Univ. Press (1960), 185-213. DICKSON, R. J. AND PERKO, L. M., Quadratic differential systems, Lockheed Research Report No. LMSC/L-56-68-1, Lockheed Palo Alto Research Lab., Feb. 1968. HART~IAN, P., “Ordinary Differential Equations,” Wiley, New York (1964). LEFSCHETZ, S., “Differential Equations: Geometric Theory,” 2nd ed. Interscience, New York (1962). ANDRONOV, A. A., LEONTOVICH, E. A., GORDON, I. I., ?.ND MAIEN, A. L., “The Qualitative Theory of Dynamical Systems of Second Order,” Nauk Publishing House, Moscow, (1966) (in Russian). equations in the plane,” MARKUS, L., “Global structure of ordinary differential Trans. Am. Math. Sot. 76, 127-148 (1954). TUNC, CHIN-CHU, Positions of limit cycles of the system dx - = co
%kX’Y”
4 yg=
&c,+~<~ . .
bnxiyk,
Chinese Math.
3, 227-284